t. tajima et al- instabilities and vortex dynamics in shear flow of magnetized plasmas

8/3/2019 T. Tajima et al- Instabilities and vortex dynamics in shear flow of magnetized plasmas

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Instabilities and vortex dynamics in shear flow of magnetized plasmasT. Tajima, W. Horton, P. J. Morrison, J. Schutkeker, T. Kamimura, K. Mima,and Y. AbeInstitute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712(Received 19 March 1990; accepted 1 November 1990)Gradient-driven instabilities and the subsequent nonlinear evolution of generated vortices insheared E X B flows are investigated for magnetized plasmas with and without gravity(magnetic curvature) and magnetic shear by using theory and implicit particle simulations. Inthe linear eigenmode analysis, the instabilities considered are the Kelvin-Helmholtz (K-H)instability and the resistive interchange instability. The presence of the shear flow can stabilizethese instabilities. The dynamics of the K-H instability and the vortex dynamics can beuniformly described by the initial flow pattern with a vorticity localization parameter E. Theobserved growth of the K-H modes is exponential in time for linearly wIsEable modes, secularfor the marginal mode, and absent until driven nonlinearly for linearly stable modes. Thedistance between two vortex centers experiences rapid merging while the angle 6 between theaxis of the vortices and the external shear flow increases. These vortices proceed toward theiroverall coalescence, while shedding small-scale vortices and waves. The main features of vortexdynamics, the nonlinear coalescence and the tilt or the rotational instabilities of vortices, areshown to be given by using a low-dimension Hamiltonian representation for interacting vortexcores in the shear flow.

1. INTRODUCTIONThe presence of shear in the flow of neutral fluids andplasmas gives rise not only to instability of the sheared layer,i.e., the Kelvin-Helmholtz (K-H) instability, but also tostabilization of other instabilities, the interchange mode[ Rayleigh-Taylor ( R-T) instability], for instance. Resis-tive-interchange-driven turbulence has been proposed as a

mechanism for the anomalous thermal transport in stellara-tors and in edge plasmas of tokamaks. Recent calculationsindicate that a strong nonuniform radial electric field cansuppress the interchange and resistive pressure-gradient-driven instabilities. The fluid dynamics of shear flows underthe influence of gravity is also important for the problem ofan imploding inertially confined plasma. In the initial phaseof implosion, short-wavelength modes are stabilized by theablative flow and relatively long-wavelength modes cangrow on an ablation surface. 3P4 arge-scale vortices excitedby the R-T instability are adiabatically compressed, andthus increase in strength during the implosion. It appearsthat the shear flows associated with large-scale-length vorti-ces suppress the short-wavelength R-T mode in the stagna-tion phase that occurs during the final phase of the implo-sion. The presence of vortices can also influence the nature ofturbulence and associated transport. In the isotropic two-dimensional (2-D) Navier-Stokes turbulence the well-known Kolmogorov power spectrum of k - 3 developedfrom space filling small-scale eddies. However, we find thatthe turbulence power spectrum changes to a steeper powerlaw in kin the presence of vertical structure in the fluid in thewave number regime on the scale of the vortices. Thus thepresence and dynamics of the vortices may strongly affectthe macroscopic behavior of turbulence.

In this work, we extend the previous work by investi-gating the shear flow effects on the gravitational instabilityand the magnetic shear effects on the K-H and R-T instabi-lities. Also, the detailed analysis of the nonlinear evolution oflarge size vortices is presented here.In magnetic confinement devices the shear flow occursat the boundary between the rotating core plasma and thewall or limiter. The magnitude and direction of the core rota-tion is determined by the strength of the nonambipolar lossrates leading to the charge-up of the plasma. The mirror oropen field line confinement system has an intrinsically fasterelectron loss rate leading to the net positive potential of sev-eral times the electron temperature. In the stellarator withstrong electron cyclotron heating there is also a dominantelectron Ioss and positive charge to the plasma. In contrast,for stellarators with neutral beam injection or ion cyclotronheating and, in general, for tokamaks, there is a net radial ionloss rate from finite ion orbits size effects and the plasmasbuild up a substantial, of order the ion temperature, negativepotential. The positive potential plasmas rotates in the iondiamagnetic direction and the negative potential plasmas inthe electron diamagnetic direction. In typical stability analy-sis the assumption is made that the rotation is sufficientlyclose to a solid body rotation and sufficiently slow that theonly effect is to Doppler shift the wave frequencies from thevalues calculated in the absence of rotation. The conditionsfor the limit of this approximation are given in Ref. 1 for therotating cylindrical plasma wi#h o*, and wlc, drift modes. Inthe presence of shear flow we can estimate the condition for astrong effect of the shear flow on a mode of growth rate yk ,Y ,wave number k,,, and the mode width Ax by the condition

k,, Ax u> yk,.938 Phys. Fluids B 3 (4), April 1991 0899-8221/91/040938-l 7$02.00 @ 1991 American Institute of Physics 938

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Here, we consider a configuration of plasma density, shearflow, and magnetic shear as shown in Fig. 1, where U = u/a.Applying this condition to the values of k, Ax, and yk for theinterchange, resistive g, and the drift wave, gives a first esti-mate for the shear flow required to reduce the growth rate.Table I shows the condition on U obtained from this crite-rion for several forms of plasma turbulence.Since the sheared velocity flow contains a source of freeenergy, one expects instability to arise from the shear flow,which it does above a critical strength. However, the formsof the eigenmodes of the K-H are sufficiently different fromthose of the interchange-drift-wave-type of instability thatthere is generally a substantial window between the stabiliz-ing effect of the shear flow on the interchange modes and theonset of the Kelvin-Helmholtz instability, as shown in somedetail for the m = 1 and 2 modes of the rotating cylinder inRef. 1.Recent experiment8 in the DIII-D tokamak show thatassociated with L (low) to H (high) confinement modetransition, there is a substantial increase in the perpendicularcomponent of the plasma flow velocity, as measured by thespectroscopic shifts of helium line radiation. No such ap-preciable change is observed in the toroidal component ofthe plasma flow velocity. Taylor et al. also report no appre-ciable change in the toroidal velocity and a substantial in-crease in the poloidal velocity with the onset of H-mode-likeplasma conditions. The abrupt change in the flow speed isinterpreted to be due to a strengthening of the radial electricfield strength. Shaing and Crume have interpreted thischange in.the radial field strength with increased nonambi-polar radial ion currents and a bifurcation to a new rota-tional equilibrium.

/Vno(0)

(b) n,(x)1 t

.,,,rFIG. 1. (a) Slab geometry showing coordinates used to describe the shearedmagneti c field B(x) and sheared flow velocity v,,(x) along with the direc-tions of Vn,, and g. (b) The pi ecewise continuous profiles of the sheared flowvelocity o,(x) and the density n,,(x).

Shaing et al. note that without consideri ng the stabilityproblem there may arise improved confinement resultingfrom the shear flow layer. Biglari et al. also discuss that theshear flow in itself may reduce the transport. A simple sin-gle-mode description of the shear flow reduction in transportis given by the convective cell island width formula

which traps plasma to form an insulating layer. Here R is thepoloidal rotation rate R = (c/rB) (da/&) and @ s the am-plitude of the vortex wave.In the present work we consider how the shear flow maystrongly modify the strength of the growth rates of the un-derlying turbulence generation from the interchange- anddrift-wave-types of instabilities in the limit of ion gyroradiussmall compared with all scale lengths, both in theory andsimulation. Theilhaber and Birdsall studied the K-H in-stability with finite Larmor radius effects fully taken intoaccount but without magnetic shear effect.A similar charge separation induced shear flow appearsin the barium ion injection in the ionosphere.* Other mag-netospheric appearancesI and astrophysical ones such asjets14 of the shear flow instability are noted. When the shearflow is sufficiently strong to dominate the stabilizing effectsof magnetic shear, the growth rate reaches a maximum forwave number k, u 1/2a, where the maximum growth rate is

YmaxCO.2 max~dv,,/dx~~O.224/~. Since the short-wave-length modes with k,,a> 1 are stable to exponential growth,vortices excited by the K-H instability extend over all theshear flow region with A, -;1, > a. When the shear flowdominates, the density and temperature fields are passivelyconvected with the fluctuations characterized byep /T, %&z/n, 6T/T.The fastest growing normal mode forms a perturbedvertical flow pattern with the axis of the vortex tilted withrespect to the flow direction, as shown by theory and simu-lation.5 The tilting of the vertical flow produces a momen-tum flux r = (0, v,,) across the shear layer. The momentumflux takes energy out of the shear and puts it into the verticalflows. Subsequently the vortices coalesce, with the dominantwavelength shifting to a multiple of the original wavelength.This shifting to longer wavelengths is a configuration spacerepresentation of the inverse cascade. Often the coalescingvortices or islands persist for long times.The effect of the electron parallel motion on stabiliza-

tion of the K-H mode is shown to reduce the maximumgrowth rate. The electron density fluctuations induced bythe electron parallel motion (V,, l ,, ) balance with ion densityfluctuations generated by the ion perpendicular motion(V,*j, ). Namely, for charge neutral currents w e haveV,*j, -I- V,, *j,, = 0.

Since j,, -7pE,, = - ne2VIIrj/mvei, , - - (d/d)X (neV,#/Bw, ) from the ion inertia current. The effect ofthe electron parallel motion is significant whenkiv& 2k:upf.

Here Y, is the electron thermal velocity, yei is the electron-ion collision frequency, 4 is the fluctuation potential, wci is

939 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima eta/ 939Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp


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TABLE I. Effect of shear flow on other instabilities.Mode Characteristics Shear flow conditionk,Ax Yk k,Axu> yr,

Rayleigh-Taylor 1Resistive g 7% =Eta i

dlnT7, z2-.dlnn r/, (threshold )Collisional drive wave

UCk;i.$-cv*ys I, =r.Dissipative trapped electron modeEZI R

the ion cyclotron frequency, ps is the hybrid ion gyroradiusgiven by ps = c(m, T, 1 2/e3, and 2u is the velocity shearlimits in the B Xl? drift velocity. For a K-H mode withk, 5 l/a the critical tilt angle 8,, as measured between the kvector and the ambient magnetic field, is given by0, z @,/a) [ (u/v, )/( a/l, ) ] I*, where I, is the mean-freepath of electrons. For 0 2 Q,, the K-H instability will bestabilized. In the case of a sheared magnetic field, the tiltangle 19-a/L, is produced by the shearing of the magneticfield, will L, the shear length. Therefore the K-H mode issignificantly stabilized when L, ~5 a/~, ) (Z,v,/ua) I*. Theratio of the parallel diffusion k f I.$/v,~ to the ion inertialacceleration k fpfk,,u is sometimes called R, as is given byR = k,,v:a4/v,,up:L s. Both the resistive g and the K-Hgrowth rates decrease with increasing R.When there exist a density g radient and a gravity forceas shown in Fig. 1, the interchange modes can be unstable.Here we use gravity to represent either the effective accelera-tion from the VB curvature drift of the ions or the accelera-tion during implosion. The maximum growth rate for thedensity gradient d In n,/dxE - l/L, and the gravityg- vf/R, where vi is the ion thermal velocity and R the ma-jor radius of tokamak, is K.When there is a shear flow with [dv,,/dxl =~/a, the in-terchange mode can be stabilized. Stabilization by the veloc-ity shear occurs when u/a > m+ This is related to thecritical Richardson number.Let us give two examples of the above instabilities. Thefirst example is an edge plasma of the TEXT tokamak. Is Theshear flow layer width 2a-0.6 cm, in which the velocity IIchanges from - 3 X ld to 3 X lo5 cm/set, the electron tem-perature T, =20 eV, the density n,- 1-2x 1012/cm3, thedensity scale ength L, = 1 cm, and the magnetic field curva-ture R k 1 m. The electron mean-free path Z, - 200 cm and

the hybrid ion gyroradiusp,r ~0.02 cm for the above param-eters The K-H modes in this case are stabilized when theshear length L, Su(2.7v,I,a/up~) 550 m. As for the in-terchange instability, the flute mode is stabilized by thestrong shear flow, since u/a = 1O/sec 2 m=: 5 X lO/sec.The second example is the Rayleigh-Taylor instabilityof the imploded laser plasma. A typical acceleration rategivesg--z/AR for the target shell thickness AR. The veloc-ity shear will be given by ac,/AR. Since the Rayleigh-Tay-lor mode growth rate is &, the stability criteria is roughlygiven by

c&AR Z [email protected] the unstable modes are limited to short wave-lengths where k 5a2/4 AR, and a k 1 will strongly stabilizethe Rayleigh-Taylo r instability.The characteristic time scale of the Kelvin-Helmholtzor interchange processes do not involve a characteristic os-cillation frequency, such as the plasma, cyclotron, or the ionacoustic frequencies, in the center of mass frame of the plas-ma. The plasma how is due to the E x B drift of the guidingcenters and the characteristic time scales are those of hydro-dynamic flows, although the elementary process is that of amagnetized plasma with long range Coulomb interactions.The effects of finite pressure density gradient and gravity,across the magnetic field and the shear flow layer, bring inthe drift wave frequencies o*, and @*pi. Thus to study thenonlinear evolution of shear flows and vortices associa tedwith the magnetized plasma through numerical simulation,time scales much longer than the plasma oscillation periodsare required. We employ the implicit particle simulationtechnique with the decentering algorithm,6 which system-atically removes the characteristic time scales and spatial

940 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima efaf. 940Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp


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length scales that are smaller than the time step At and spacescales Ax chosen for the space-time grid. The filtering meth-od has been shown to preserve the accuracy for the low-frequency (w At < 1) dynamics.In the present paper we investigate the nonlinear evolu-tions of the Kelvin-Helmholtz and interchange instabilitiesas an initial value problem through particle simulation, incontrast to the previous work,5 where the shear flow wasexternally fixed with an imposed driver, as would arise from

nonambipolar losses in the background plasma. Namely, inthe present simulation, we assume that a space-charge sepa-ration exi sts initially, which produces an initial E X B shearflow. Note that any processes that i nduce charge separationhave not been included in the simulation. The seculargrowth and decay of the marginally stable normal modes arealso studied. After the linear stage of exponential growth ofthe primary normal modes, the growth of secondary modescan be nonlinearly triggered.In order to systematically explore the parametric de-pendence of the development in the nonlinear stage, we iso-late the evolution of vortex coalescence and associated pro-cesses aused by vortex formation, which in turn is due to theK-H instability and its nonlinear evolution. To investigatethe second stage, the system is initiated from the secondaryequilibrium of a chain of finite-amplitude vortices. Thechain of vortices is unstable against the coalescence modeand against the tilt or rotational mode. In this nonlinearregime the growth of coalescence and tilt modes are nonlin-ear instabilities showing the finite time singularity like(t, -t)-fortimesta,v = u(x/a), Ixlca,- u, x-c -a.

The configuration is schematically shown in Figs. 1 a) and1 b) . Except for especially indicated cases, we consider theprevious plasma configuration. Also, gravity is applied inthe x direction, which destab ilizes (stabili zes) the inter-change mode for r = g/L,, ? 0.In the case of low-frequency modes with relatively longwavelengths /2>&, and pi, where &,, and pi is the elec-tron Debye length and the ion Larmor radius, the wave dy-namics can be analyzed by fluid equation for electron andion, and the condition of charge neutrality can be assumed.Namely,

n, = n, = n. (1)From the electron equation of motion along B, we obtain theequation for the parallel electron currentj,, ,me(& -I- V.P)io = - ne2V I14+ eV,,p, - mev,j,, --O, (2)where vE is the EXB drift, 4 is the electrostatic potentialperturbation, p, is the electron pressure, and vei is the effec-tive electron collision frequency. Using the electron equationof continuity and Eq. (2), we obtain($ +v.d,)rr,~ll~.ill

=--1_(-noeV~4+Vfpe).m vei (3)Assuming T, constant and nc = n,(x) ( 1 + s,), Eq. (3) isrewritten as

(P+vE*V)Re = -~$+$vp, -$), (4)where $ = eqS/T,, L,, = (d In n,/dx( - , c, = ,,/m,and the ion hybrid gyroradius ps = c,/w,+, where w,, is theion cyclotron frequency. In Eq. (4), nonlinearities otherthan the nonlinear polarization drift are neglected.From the ion equation of motion, we obtain vi,, the iondrift velocity perpendicula r to the magneti c field

Vi1 =vE +Vg d-v, fvpt (5)where

VE = C[ZXV($ + $5,1/B&, (6)vg = - (g/w,, 19, (7)vd = [ CgXVp, )/neB,]c, (8)

andVP - $+ (7, +v,)-v, >v, (40+ 4)C. (9)

Here &, is a background plasma potential. Note here thatwhen finite ion Larmor radius is included, the convectivederivative of Eq. (9) may be replaced by ( vE + vd + v, )-V,as discussed in Refs. 18 and 19. The ion equation of continu-ity and Eq. (5) imply

941 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima et al. 941Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp


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( & + (v, + vgw, >nj -p;v,- nt[ (&+(vE+vg)% >V,ta+so+o. (10)

Setting nt = b(x) (1 -k 2, ), USing i, = [ (iXV$)/B,lc,and definingc 4% gJO=------,B, ax (11)@ciwhich is the ambient ion flow veloci ty, Eq. ( 10) yields,( g+ v,+vg)vI>(ii - -$V, lnoV, 4n0 >

= 2 ,,a? w a$ pf a$----PJQy L, ay tpav' (12)where the prime indicates d /dx. Equations (4), ( 12 ) , andthe charge neutrality condition of Eq. ( 1) are our basic equa-tions. Note that only the dominant nonlinearity is retained inEq. (12),asinEq. (4). WhenwelinearizeEqs. (4) and (12)and set v, = 0 and V,, = 0, Eqs. ( 16) and ( 12) can be re-duced to Eq. (31) of Ref. 11 and Eq. (7.17) in Chap. 7 ofMikhailovski and to the Rosenbluth-Simon equation.In the absence of the gravitational drift velocity and fora uniform vE the coupled equations (4) and ( 12) reduce tothe well-known Hasegawa-Wakatani equations describingthe collisional drift wave. In the low collisionality-strongshear limit k iv: > vei [wk 1 he density is forced to be close tothe local Boltzmann distribution and the equations reduce tothe single dissipative equation2 often used to study driftwave turbulence. Including the gravitational accelerationg/L,, gives the resistive g mode for the collision dominatedplasma and an additional stabilizing or destabilizing effect tothe drift wave in the weak collisionahty regime.Let us look at a linearized wave equation for a mode thatvaries as &(x)exp[ - iot + iky + ik,, (x)2], wherek,, (x) = kx/L,. Eliminating fi = Fz,= ii, from Eqs. (4) and(12), we obtain;&!-(no~) = (p;k2 p:k-j;Ltt)

- (ku, + ik f;Q ) (kp,c,/L, )(~-kvol(~-kv~~ fik&)+ ik 4,w - kv,, f ik f D,, >A (13)

where D,, = v~/v,, is the parallel electron diffusion coeffi-cient and uEo = c d&/ax B. Equation ( 13) includes variousMHD instabilities driven by gravity, shear flow, and densitygradient, which correspond to interchange instabil ity, Kel-vin-Helmholtz instability, resistive g mode, and drift waveinstabilities. In the following discussion, shear flow stabiliza-tion of the interchange mode is investigated.We derive dispersion relations for the following twocases.Case 1 s the discontinuous density step: b -0, no = It,,for x > 0, no = n2 for x < 0, U/a = const as a-+ 00 and nomagnetic shear (L .~ t 00 ). Case 2 is the smooth density

chaws: a = b and n,(x) = no exp(x/l,) for 1x1 O,

and$=Dekx, x


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v = k [II,, , and we assume Re K, < 0 and Re ~~ > 0. Thejump conditions are 4?(a) + t4t IA = B + CeJ:,D = Be - *I + C, (21)

andK; (a)A - K,(Q)B - KZ(Q)Ceti2

=-- ku/a Aw+ ku - kv,

K,( -a)Be-l+~*( -Q)C-K;( -a)D (22)= + ku/a D*- ku - kv,

Here $,,Z = .f- o~,,2 (x)dx. Equations (21) and (22) yieldthe dispersion relationK; (Q) -K,(Q) - ku/aw - k(u + up>

x K2( -Q) -K;( -Q) + ku/aw+k(u-v,) >ku/a- K; (a) -K*(Q) - w - ku - ku, >

x K,(( -a) -K; ( -a) + ku/aw + ku - ku, >xe-+&=o , (23)whereK:(Q) =K,(Q) 1 ,L,,=II and K;( -Q> =K2( -a)l,,L,,=o.This dispersion relation i ncludes both the K-H instabilityand the interchange or the resistive pressure-gradient-driveninstability.

1. Case 2(a). Magnetic shear stabilization of K-HinstabilityThe density gradient l/L, and us are set to zero in Eq.(23) to obtain

(ku/a)*co* k *u* exp( - fy, dx)+ 2q+s >( ku/a2q- -- w + ku > = 0, (24)where qr and qi are a real and imaginary part of q, respective-ly, and q * = q( f a). As for the growing mode, we assumew = iy is pure imaginary. This assumption is justified sincethe imaginary part of the left-hand side of Eq. (24) is pro-portional to the real part of w, which can be set to zero.Equation (24) is rewritten by keeping in mind4+ = qt =q,(Q) t iq,(Q) as fOllOWS:

1 - 4Qcl,(Q)-exp( -2JIoq,dx)+4a(q:+qf)] =Q (25)

which yields

X[exp(2j-~aq,+-4alq+where

qr(a) = [(s2+ t*)* +sl/*/Jz,

(26)

(27)C?,(a) = - [(S* + t*)* - s]*/Jz, (28)

ands=k*+ v(v + yvp5 =k2+ d/P?(v + y)* + k*U* y2 + k*u* (29)t= vku/p: vku/p:(v+y)*+k*u*=yZ+k*u** (30)

Since qi (a) in Eq. (26) is negative, the K-H mode is unsta-ble whenexp( -2JIoq,dx)- [2q,(a)a- l]*--4qf(a)a>O.

(31)Roughly speaking, the maximum growth rate is atq, (a)~ c ka=J and the threshold with respect to the parallelwave number is

ak,, (Q) < 0.3OJm. (32)When the shear scale length L, is shorter than

L, = 1.65 V,i,Q/l&) I*,all of the K-H mode will be stabilized.

(33)

2. Case 2(b)Without shear flow and magnetic shear, the dispersionrelation (24) gives the growth rate of the interchange insta-bility for a finite density grad ient. Equation (23 ) reduces totanh(2aq) = - 2kq/(k* + q*), (34)

where we assume kL n E L,/p, >>1 and approximateK1.2 = -+ 4,q = k [ 1 + (g/L,/(w - kv,)w)] I*. (35)

Setting q = iz, Eq. (34) becomesf(z) = tan(2az) = 2kz/@ - k*). (36)Solutions of the dispersion relation correspond to the crosspoints of Fig. 2. As seen in Fig. 2, the solutions of /z/k ) < 1are approximately given by

ZQ-(r/2)l, I= + 1, + 2 ,..., (37)which yields the frequency-=2+,/m. (38)Therefore the models unstable when

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FIG. 2. Graphical location of roots of the eigenvalue problem given by in-tersection oflc(z) = tan(2az) and the right side of Eq. (37).

Since g,cf/R for the magnetic field curvature R andR 2 L,, the mode is unstable up to k Zz /p,.3. Case 2(c)Finally, we briefly discuss the velocity shear effects onthe interchange instability in a finite density gradient. By aprocess similar to the derivation of Eq. (36) from Eq. (23),we rewrite Eq. (23) wi thout any approximation to obtain

tan( -iJIoq(x)dx)=$,wheres=q+q- +(K- -&)(G +$--)

ku K, + 1/2L,-- a w - kv, + kuku K- - l/2&,+- k %*/a*a o- kv, - ku ,

T= --( 2;n +K++ -$~+,*~jq?*ku q+-- Q w- kv, + ku +& q-a w - kv, - ku

K + = [F + iv/p: (0 F ku - iv) ] I*.Here we used relations

K1.2 = f I/=, t- qtand

4i =q( fQ).

(39)

(40)

141)(42)

(43)

(4.4)Since the dispersion relation of Eq. (39) is similar to Eq.( 36)) the solution of Eq. ( 39) is approximately given by

s0 q(x)dx=i:l+S (Z=O,& l,f2 ,... ), (45)-a

which corresponds to Eq. (37), where S is a phase factor ofthe order unity. The WKB approximation that was used toderive Eq. ( 19) is valid when the integer I is sufficientlylarge. Assuming kL, $1, q(x) is approximated by

q(x)= k--( (ku, - iv) kc,/L,p,(o - kv,) (a - kvEO+ iv)-I- iv/p:w-kv,, -f-iv (461

There are two resonances in Eq. (44), which are located atX,1 = [b - kv,)/ku]a

andX rt = [(w + iv)/ku]a.When 1~1 kv,, i.e., the magnetic shear is small enough, thedistance between the two resonance points is au,/[u 1.Evalu-ating q(x) at the center of the two resonances, namely,q( !$+o -2 k2v;4W2)2,

the growth rate obtained from Eq. (46) is roughly evaluatedto be

Therefore if V(481

the interchange mode is stabilized by the shear flow. Notethat Z? = fL,/gl ( U/Q)* is the Richardson number. Accord-ing to Chandrasekhar s textbook,* 5? < 1 s given as the sta-bilization condition of K-H instability as for the gravity inthe stable direction.In unmagnetized plasmas the linearized equation, in-cluding equilibrium shear flow and gravity, is derived fromthe inviscid and incompressible fluid equations. The similarequation has been derived by Chandrasekhar,** in which theeffect of gravity on the K-H instability is discussed. Theresult is

k =P?PO + W + kW, /po(co kv,) o - kv, w - kv, > vx.(49)As for the shear flow stabilization of the Rayleigh-Taylorinstability, the stabilization condition for the configurationof case 1 is exactly the same as that given by Eq. ( 18). Thecriterion of Eq. ( 48 ) for case 2 is also applicable to Eq. (49 ) ,III. INlTiAL VALUE SIMULATION OF SHEAR F LOWINSTABILITIES

The static uniform magnetic field B, is now in the zdirection only. The initial ion density is uniform, n; = n,, inthe x-y plane. The pl asma is encased n a metallic box in the xdirection with 4(x = + LJ2) = 0 and periodic in theydi-rection for most of the computer experiments we present,unless otherwise specified. The particle velocities are reflect-ed at the x boundaries. We load the electrons with a densitygiven by944 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima etal. 944



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n, (x,t = 0) = no + An,/cosh2(kg) , (50)where typically An, = 0. In, and k, = l/a, with a being theshear layer width of the E X B flow produced by the chargeseparation py = e[ n, ( CO - n,] . The initial flow of theplasma produced by the charge or vorticity layer given in Eq.(50) is

u,(x) = (4an,,ec/k&,) (An,/n,)tanh(k+)= u. tanh (X/Q). (51)

Although we vary parameters over a wide range, the typicalset of parameters are as follows: the numbers of the gridpoints in the x and y directions L, = L, = 64, the numbersof particles in the x and y directions N, = N,, = 192, theelectron cyclotron frequency o,, = SOW,,with oPe being theelectron plasma frequency, the ion-to-electron mass ratioM/m = 1600, the shear width a = k 0 = 6A with A beingthe unit grid separation, the electron and ion Debye lengthsperpendicular to the external magnetic field direction/2,, = /zm = 0, the electron and ion Larmor radiipe = p, = 0, and the simulation time step At = 2OOw, . (Inother words, particles are loaded cold with no thermal veloc-ities in the directions perpendicular to the B field. No tem-perature development is seen well beyond the linear stage.The shear width is input as CI= 5A, which gives rise to theeffective shear width of 6 as a result of the finite size particleeffect.) Note that (W,i/W,i)2 = (m,/me)(u,,/u,,)* = 4,which is orders of magnitude smaller than that of the usualfusion plasmas. Instead of assigning a uniform weight of uni-ty to an individual particle, the weight of the particle is deter-mined by the fraction n, (x)/n, dependent upon its initiallocation. The weight of the particle in the simulation is notchanged throughout the run. In the reference simulations wechoose Andn, = 0.1, the size of particles LI, = a,, = 3A, andthe decentering parameter yi = 3/e= 0.1.The linear theory*5,23 for the hyperbolic tangent profileof Eq. ( 5 1) gives that the Kelvin-Helmholtz mode is unsta-ble for the wave numbers k,,, satisfying

k,a < 1, (52)where k, = 2rrm/L, and m is the mode number in the ydirection. Figure 3 shows the electric potential ]@I2 as a

FIG. 3. Evoluti on of the electri c potential Qf,, (t) for modes m = 1,2,3,4 forthe reference parameters in Sec. III.

function of time for each mode (m = l-4). Notice that be-cause of the lack of noise in the implicit particle code, a largenumber of decades of exponential growth of the instability isobservable. After a short period of time modes with m = 1and 2 grow exponentially in time, while modes m = 3 and 4do not grow until well into the nonlinear staget-5x 104wP; = 31a/u,. Here recall that the thresholdmode number m, = L,/a = y and m = I,2 are supposed tobe linearly unstable and m>3 are stable. This is in agreementwith simulation in Fig. 3. For these simulation parametersbru,(x=L,)At/A is equal to 0.75. The m =2 modeshows a slight oscillatory feature, as seen neart = 8x 104wP; = SOa/u,. The modes with larger modenumbers are triggered unstable after the amplitude of thelinearly unstable modes becomes high enough and the vorti-ces of these modes begin to interfere or overlap aroundt-5x10%,.Figure 4 exhibits a typical particle plot and the corre-sponding electrostatic potential contours. For clarity, onlyparticles with initial velocity uu > 0 on the left half at t = 0are shown. Figure 4 is at t = 1 X 10~~; = 62.5a/u,. In Fig.5 we show the measured and theoretical growth rates of themodes. Figure 5 also shows the measured growth rate for thecase when the magnetic field B. is tilted toward they direc-tion from the z direction by angle 0 = 0.010 rad. Note thatthis is less han the critical angle (m,/M, ) * = 0.025 rad. Inthis case the flow is still unstable although the magnitude ofthe growth rate is reduced by a factor of one order of magni-tude and the unstable wave number increases, as is charac-teristic of drift-wave-like modes. On the other hand, whenwe tilt the magnetic field away from the z axis by0 = 3 X 10 - *, the system is stable. The electron thermalspeed uth is taken to be O.O5w,,A in the tilted B field runswhere electrons can move along the magnetic field line,while the thermal velocity perpendicular to B remains zero.Thus k,, u,,, = 9.8 X 10-4m~pe or the 8 = 10 - * case, wherem is the mode number in they di rection. Thus y,,,,, < k,, u,,,where yrnax= 1.25 X 10 -4~pe = 0.2Ou,/a for the K-Hmode (in Fig. 5). For 8 = 3 X 10 - *,k,,u,,,g.F-. The mea-sured maximum growth rate for 0 = 10 - * is0.26 x 10 - 4~pe, about one-fifth of the 13 0 case. When thenormal mode is marginally stable, we find that the growth of

FIG. 4. (a) Contour plot of the electrostati c potential at the critical timeshown in (b). (b) Position of particles in the (x-y) plane with initial fluidvelocities o,


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this amounts to the following initial electron density:a, (4.Y) = no An,(l -2)[cosh(kfi) + E cos(k#) I2 (53)

IIk o-Y

FIG. 5. Compariso n of the measured growth rates with theoretical growthrate for the collisionless piecewise linear slab flow. For the * data the tiltangle 6 of the magnetic field is zero and for the A data the tilt angle is0 = O.OlO/rad.

the mode becomes secular, as seen in Fig. 6. The electricpotential resul ting from the marginally unstable mode in-crease linearly with time.As has been shown in theearlier driven simulations, thestage at which the nonlinear triggering of other modenumbers sets in is coincident with the development of a vor-tex chain. In order to better control the study of the nonlin-ear problem of vortex evolution, our approach here is toseparate the linear and nonlinear stages. We idealize theproblem by starting from the secondary equilibrium of a vor-tex chain.IV. NONLINEAR EVOLUTION OF VORTICESA. Comparison with implicit particle simulation

In this section we initialize the simulation near the Stu-art-Kelvin cats eye equilib rium.24*25 In the plasma context

FIG. 6. Secular growth of the potentia1 with a linear in crease in time for amarginally stable mode.

where O


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noise due to numerical truncation that results upon loadingthe particles, as there is no noise associated with the particlemotion perpendicular to B with the initialization and thesubsequent decentering algorithm. I6 Att = 4.2 X 104wP; = 26&c the contour lines show a recon-nection o f the flow lines, reminiscent of the tearing instabil-ity of magneti c islands, which yields vortices (islands) withsmaller wavelengths (m = 4). Note the original vortices hadmode number m = 2. At a later time the smaller inducedislands (m = 4) are absorbed by the original islands(m = 2). At a later time the smaller induced islands(m = 4) are absorbed by the original islands (m = 2). InFigs. 7(c) and 7(f) at time t = 6.6X 104wP; = 41a/v,, weobserve that the larger original vortices coalesce into onevortex with m = 1 in the direction of the exterior flow. Wealso see the vortex tilts in the clockwise sense as a result ofthe ambient external flow, whi ch is downward on the rightside of the vortex and upward on the left side.During the coalescence process the perturbed electro-static potential energy grows exponentially in time as shownin Fig. 8 (a). Figure 8 summarizes the growth of the electro-static energy for cases with various values of E. The satura-tion of this energy in Fig. 8(a) occurs shortly after a com-plete coalescence.As we raise the value of E, with other parameters beingfixed as before, the growth rate of the electrostatic energyincreases, as shown in Fig. 9. This figure will be further dis-cussed in Sec. V. In frames (b)-(d) of Fig. 8 we also observethat a slight bump develops in the middle of the otherwiseexponential growth phase. In particular, Fig. 8(d) shows afaster than exponential growth in the early stage, while set-tling into a nearly exponential growth later. This indicates atransient growth that is faster than exponential growth forE > Lt ~0.5. Another feature to be noticed in Fig. 8 is theamplitude oscillations after the coalescence. These are asso-ciated with the ringing of the vortex shape. This is reminis-cent of the coalescence process of magnetic islands, althoughany parallelism of the coalescence of vortices in the presentinvestigation with that of the magnetic island coalescence isperhaps fortuitous since the dynamical equations are ratherdifferent. Some conspicuous differences in the governing

,~~~~1

0 kdp- twp- 9x104FIG. 8. Evolution of the electrostatic potential energy from the nonlinearisland chain as a function of increasing vortex strength e.

L IOo I I I 1 I I I 1 I0.2 0.4 0.6 08 IOE--FIG. 9. Plot of normalized linear growth rate G vs E: X represents a datapoint extracted from the simulation results, 8 is a data point where it isdifficult to ascertain the growth rate because he growth is faster than expo-nential. The dashed curve is the linear theory of Ref. 25; the solid curve isthat derived from the localized vortex model.

physics include (i) the vortex dynamics is described by thesingle field 4, while the magnetic island dynamics requires atleast two fields 4 and the z component of vector potential A,;(ii) consequently, there exists a magnetic repulsive forceupon magnetic island coalescence, while in the vortex dy-namics there is none; and (iii) the magnetic flux conserva-tion inhibits the reconnection of the magnetic flow lines. Onthe other hand, the presence of the Kelvin-Helmholtz insta-bility and the coalescence nstability for the vortex dynamicsis similar to the presence of the tearing instability and themagnetic coalescence instability, except for the frozen fluxconstraint that d$/dt = vV*$+O. The importance of theflux conservation constraint is easily seen n the formulas forthe linear growth rates where yk-- k, hvv butya~k:/3~~~.In light of the above the actual dynamics of the tearingmode is, in principle, different from that for vortex coales-cence dynamics. Ideas used to study the tearing, however,can be used to measure the vortex dynamics observed here.One measure is the vorticity difference between the original0 point and the innermost X point (e.g., between A and B inFig. 7 and Fig. 12), and another measure is the vorticitydifference between the original 0 point and the outermost Xpoint (e.g., between A and C in Fig. 7 and Fig. 12). Theformer measure of vorticity is indicated by circles and thelatter measure by crosses in Fig. 10, for various E cases. Bydefinition of the former, the measure of vorticity vanisheswhen the two vortices complete their coalescence. CompareFig. 10 with Fig. 8. On the other hand, the latter measure ofvorticity may or may not vanish. In small E runs [Figs. lO( a)and 10(b) 1, we see that it decreases until a certain point(t- 5 X 104wP; = 38.6a/v,) and then begins to increase.This manifests itself in a larger vortex att = 6.6 x 104wP; = 5 a/v, [e.g., Fig. 7(c) ] than the origi-nal vortex. In larger E experiments [Figs. 10(c) and 10(d) ]the two measures depart, but both measures decrease, or atleast not increase, even well after the coalescence.

Let us further examine the cases with E = 0.3 and947 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima et /. 947



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I (ok=008

1:1:11T_1.::;:~~:::~:~~

0 +wp 9xtoo twp - 9

=!

FIG. 10. A measur eof the vorticity in the trapped coherent structures. Thevorticity difference between the original 0 point to the innermos t X point isgiven by the circles (0) and vorticity difference between the original 0point and the outermost X point is given by the crosses x ).

E = 0.6, which are shown in Figs. 11 and 12, respectively.Figure 11 c) shows skewness of each vortex, as well as thetilt of the axis of the two vortex centers, meas ured in thenegati ve direction. This is similar to the prediction by Liu etal. and one found in the simulation.5 A more pronoun cedtilt may be seen n Fig 12. The rotation of the axis continueseven after the completion of the coalescence.Thus we findthat the chain of vortices is unstabl e against the tilt or rota-

tw, - 3.6 x O -FIG. 11. Potential contour s (a)-(c) and particle plots (d)-(f) for theperi-od-doubling coalescence f the m = 2, E = 0.30 island chain. The skewnessof each vortex as the axis between two vortex centers rotates during coales-cence.948 Phys. Fluids B, Vol. 3, No. 4, April 1991

6 20.6064

64 twp=5.6xId two=6r103

64

FIG. 12. As in Fig. 11, but a caseof e = 0.6 with stronger tilting, where therotation continues after coalescence.

tional instability. In Figs. 12(a)-12 (d) we obse rve herota-tion of the axis that connects the two 0 points as they ap-proach each other. Even after the coalescence he rotationcontinues. At the same time the overshooting oscillation(squashing of the droplet) continues. In this particular case,where E = 0.6, during the course of these droplet vibrations,fission of the vortex occurs, as seen n Fig. I2 (d) . Observe nFig. 12 that as the m = 2 vortices coal esce into an m = 1vortex, much smaller-scale vortices spring up. As the energyinversely cascades rom the m = 2 vortices to the m = 1vor-tex, the enstrophy cascade s rom m = 2 to higher ms,sinceboth the overall enstrophy and energy are conserved. Fromthe distribution of particles in plots in Figs. 7, 11 , and 12, wenote that e ven when the potential contours show fairly co-herent patterns, the particles are strongly mixing in complexstructures.In Fig. 13 the negative of the rotational angle of thevortex-vortex axis as a function of time is measured. As cssincreased, so is the growth of the angle. The rate of increaseof the angle before 8 = 180 s found to be faster than exp o-

Tajima et& 948



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64 t=o (a)

00 2.4 4.8 7.2 0 X- 64+a.--

FIG. 13. Simulation measurements for the rotation angle as a function oftime 0(r) for labeled values of E.

nential. Following the terminology of magnetic coalescencethis growth is called explosive growth. This explosive in-crease of 8 saturates at or near 8 = 180. In some cases 0stays around 180 after it reaches this position. In othercases, after a brief pause at 8 = 180 the angle again in-creases. The higher value of 8, the stronger is this tendencyfor continued rotation. Figure 14 displays the distance be-tween the two 0 points as a function of time. Once again thisdistance Sr(t) = J-z also grows faster than expo-nential during the coalescence, indicating the explosive na-ture of the transients in the coalescence process. Note thatdifferent from 0 in Fig. 13, the dependence of Sr as a functionof E is not monotonically increasing. Also noted is that theincrease of 6r as a function of time is sometimes not mono-tonic.Figure 15 shows the potential 4 at t = 0 and a later timeat which point a nearly ?r/2 rotation of the axis of a pair ofvortices is realized for the E = 0.3 and E = 0.7 cases. Theevolution will be later compared wi th the theoretical modelin Sec. IV B.B. Localized vortex model

We now present techniques for analytically modelingvortex simulation results, such as those presented in Sets. IIIand IV A, by simple few degree-of-freedom Hamiltonian

3(

+sr

I-

O t0 twp + 6~10~-J

FIG. 14. The distance between two 0 points as a function of time E+(t), asmeasured in the simulation for the labeled values of E.

64 t=o (c)

tY

64

0 x-+ 64FIG. 15. Equipotential contours obtained from the simulation: (a)E = 0.95, t = 0; (b) E = 0.95 at a later time; (c) E = 0.7, t = 0; (d) 6 = 0.7at a later time.

systems. This is a plausible goal, as the system is nearly dissi-pationless and low wave number modes dominate the dy-namics. The specific application of these techniques to thenear cats eye simulations are considered. The goal here be-ing to elucidate the dynamical mechanisms that are activeduring the coalescence or tilt instabilities.The simulation results suggest that there is a large tem-poral regime where the two localized vortices of the initialcondition remain isolated and maintain their integrity whilemoving. This suggests a few degree-of- freedom models forthe flow, composed of localized interacting vortices, perhapssubject to an external field.Since the computational studies of this paper are period-ic in they direction, but not the x direction, these boundaryconditions must be incorporated into a model of the dynam-ics. The x boundary condition is straightforward, since to alarge degree the motion of the localized vortices is farenough removed from the boundaries for us to assume- 00


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y,,,(f) = ye(t) + Bn-n/k,,~rn+i(t) =~,(t) -i-4rn/k,,x*n (t) = x,(t),%I+,(0 =x,(t),

(55)

where n = + 1, + 2, + 3 ... .In general , suppose that the velocity at vortex n locatedat x,- = (x, ,y, ), resulting from vortex m located at x, , isgiven by

V nm = V(x, - x,). (56)Upon superposing, the total velocity at the location of vortex0 is given by

v,= -g V(x,-x,1.m= --r

rn#OSimilarly, for vortex 1 ,

v,= 2 V(x,-xx,).m= --a:

(58)mp I

Suppose that V is derivable from a streamfunction, definedbyV(x,y) = ~XV?NX,V), (59)

where $ is even in both of its arguments,?KcY) = $( - KY) = $(x, -VI. (60)

Making use of these symmetry conditions and the periodic-ity constraints of Eq. ( 55) yields the following for Eqs. (57)and (58):m= --no (

4v7mv()= 2 ;xv+ x,---xl,yo- y1 -- ko > (61)v, = iin= -* PXV$(X, - XO,Yl -Y. - T). 0We ignore the self-interaction of the vortices and assumethey move with the local flow. This yields the followingHamiltonian equations of motion:

dHi;=--, * dHay; yt=z' i= O,l, (62)whereH(x,-x,,yo-y,)= 2 t+o-xlYYo-Yl -g.,?I= -0e ( (63)The form of the Hamiltonian of Eq. (63) suggests the intro-duction of the center-of-mass coordinates, defined by

$5=x0--XI, 71=Yo--YI, (641- .I2-=x, +x0, 3=y, +y(yThe coordinates (F, 9) remain fixed in time while ({,v)satisfy

& 2Q77) ?j=~({@a7 ac (65)In order to effect the modeling, the function H({,r])remains to be determined. This can be achieved in two ways:

first, a model H can be obtained directly by tracking the

relative motion of the vortices. Since for this system physicalspace (&f;rl) is the phase space, and since trajectories lie oncurves of constant H, onecan attempt to fit H to the simula-tion output. Alternatively the vortex-vortex interaction $can be postulated, perhaps, by examination of the vortexshape. Knowing @, he sum of Eq. (63 ) must be evaluated inorder to determine the dynamics.Sometimes the sum of Eq. (63) can be evaluated inclosed form; for exampIe, in the case of point vortices where$(@I) = q. InG + $1, (661

this is the case. Defining z = (kd4)(77 + y), it is evidentthatJ - ai! jag - ko~o i 1.a7j af 2 mz-oc.z-mmlT (67)

The sum of Eq. (67) is the Mittag-Leffler expansion forcot z, which impliescosh( k,{ /2) - cos(k,T/2) > (68)

and thus from the first equality of (67) we obtain, to withinan additive constant,H(&q) = z,& n[cosh(k,CfZ) - cos(k,q/2) 3. (69)The summation performed above is related to the solvedclassical problem of obtaining the velocity field as a result ofan infinite chain of point vortices,26 but here the context isdifferent, in that H determines the dynamics subject to theconstraints (55). Here H is not the streamfunction. Belowwe will construct the streamfunction as a function of time.We propose the following form for H in the case where Ecan differ from unity:

H(S,q;c) = 11, n [ $;(E + cash k,c) - $ cos( k,q/2) ] .(701Arguments in favor of this seemingly obscure choice willshortly be given, but first consider the inverse problem forobtaining $( ~,v;E). Suppose $ has the form of the point vor-tex interaction, except isotropy is broken by warping the xdependence of the interaction, i.e.,

ljrG77;~) = fjo lnlf2(C) + ~~1, (71)where the functionf(g) is yet to be determined. Substitutionoff for 6 in our treatment of the point vortex case yields

aH koh-= sin ( k,q/2 1877 2 cosh(k,f/2) - cos(k,q/2) (72)

aff ko@o-= sinh (k,f/2)af 2 cosh(k,f/2) - cos(k,v/2) (73)Equations (72) and (73) implyH&W) = Jt, In[cosh(k,f/2) - cos(k,q/2)] -I-const.(74)Choosing the functionfas follows:

cosh(k,f) = (l/e)cosh(k$), (751results in, apart from an unimportan t additive constant, theH given by Eq. (70).There are several favorable attributes that lead one tochoose the H of Eq. (70). To begin with, it is a continuous

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deformation away from the case where E = 1, a case wherethe model agrees with the simulation [cf. Eq. (69) 1. Thus atleast near E = 1 we expect reasonable agreement. Inspectionof Eq. (75) reveals that for E# 1 the logarithmic singularityat CJ 77= 0 has been eliminated. Any distributed vorticityarrangement will have this feature. Also, the anisotropy in-troduced is in agreement with that observed in the simula-tion (cf. Fig. 15). In particular, cY$/&$ II =e dominatesd$/&7] I = o. A convincing argument in favor of the choice ofEq. (70) is that it leads to a #(x,y,t), which, as we shall see,looks like the simulation. We emphasize, however, that thischoice is not unique and only qualitative agreement issought.The construction of d(x,y,t) requires that the contribu-tions from the double infinity of vortices be summed at eachmoment of time. We conclude thatW,YJ) = 2

,n= -0c[ d(x - XOYY - Y o - 9)

045-m+q x-X,,Y-Y, -7( *0 >I (76)

Without loss of generality the arbitrary constant can bedropped. Assuming.P=yY=o,x0 = g, XI = -g, (77)Yo= 4% Yl = - pi%

yieldsQl(x,YJ;E)=H[x -&t)/2,y- 7(t)/2;el

+H[x+&W2,y+ 7j(t)/2;~1. (78)At time t = 0, Eq. (78) should represent the cats eye initialcondition. Using g( t = 0) = 0, q( t = 0) = 2r/k,, Eq. (70)implies

qVx,y,O;~) = $. In [cash k,,x + E cos kg],the equili brium state desired. At later times,

(79)

&x,y,t;e) = qbon(2[Jm- J; ..p(y - Y))

x[J~- J; cos$(y + Y))) . (80)

Now consider the comparison of the linear theory of thelocalized vortex model with the simulation. As noted above,6 = 0 and 7 = 2n-/k, correspond to the cats eye equilibri-um, but, also, these correspond to dynamical equilibrium ofthe localized vortex model. This is evident upon differentiat-ing Eq. (70). Moreover, expanding H to second order yieldsthe following Hamiltonian for the linearized dynamics:h(w1?;E) = G$o@

8[-+&1-&6#+gy* E(81)

Thus the linear growth rate based on this model, normalizedby k i q&/4, s given byp = 234E4/(1 + E) I+( Jr+E + $G,. (82)AFor E = 1, y = 4, the classical result for the max imumgrowth rate of a row of point vortices. The localized vortexmodel selects the maximum because this is the only motionallowed by the periodicity constraints of Eqs. (55). Exami-nation of Fig. 16 reveals that the simulation is in agreement

in this limit. As e-+0, p-234&4. This vanishing growthrate is in disagreement with the simulation; not a surprisingresult since the assumption of localization of the vorticesbreaks down. Because of E14 behavior this disagreement isconfined to O(E 5 0.2. The theory is in reasonable agreementfor a large range of e away from unity. In Fig. 16 we have alsoplotted the results of Ref. 25, where the linear eigenvalueproblem for the cats eye equilibrium was solved numerical-ly. Observe that for e-O.3 there appears to be a transitionfrom localized vortex behavio: to what we refer to as K-Hbehavior, i.e., sustainment of y. This is further evidenced inFig. 10. On the other hand, the theory2 by Pierrehumbertand Windall is correct at E = 0 and agrees reasonably withsimulation for E < 0.3, but is unable to converge beyond.Now consider the nonlinear behavior. Since H is con-served, one can obtain the orbits in physical space by simplyplotting surfaces of constant H. In Fig. 17 we have done sofor different values of E. This figure shows that the energysurface decreases n width as E decreases from 1 to 0. In thesimulation it was observed that for small values of E the in-

ko77

kc& kc&FIG. 16. Contours of the vortex-vortex interaction otential$ given yEqs. (75) and (79),forthecases (a) ~=0.05, (b) ~=0.30, (c) l =0.70,and (d) e = 1.0.

951 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima eta/. 951Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp


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kO6 kOCFIG. 17. Contours of constant interaction Hamiltonian, H, for (a)~~0.05, (b) e-0.30, (c) E=0.70,and (d) E= 1.0.

stability that occu rs is of the pairing or coal escence ype, asshown in Fig. 10. For values of E near unity the instabilitythat occ urs is of the tilt or rotational type. Figure 17 explai nsthis tende ncy with E. For all finite values of E the vorticesapproach ea ch other and mo ve transverse to each other. Forsmall E he later motion decreaseswith the width of the ener-gy surface and we observe the predominant coalescence.In Fig. 18(b) we show the results of integrating Eqs.(65). Here 8(t) and Sr(t) = r(0) - r(t) for various valuesof e are given. Many features of the correspondi ng quantitiesfor the simulation Figs. 13 and 14 are reproduced, as seen nFig. 18(a). Th e initial conditions here were chosen near theseparatrix, either just inside or outside. In the casewhere theinitial condition is just inside, B can increase beyond 180.This behavi or is seen in Fig. 18(a) for the case whereE = 0.95 and in Fig. 19(b) for the case with E = 0.3. Whenthe initial condition is outside the separatrix, in the localizedvortex model, 8 can only appro ach 180. This behavior isindicated in Fig. 18(a) by the flat spot near 8 = 180 andshown in Fig. 18(b) for the case where E = 0.3. Similarly,Figs. 1 4 and 18 for SP( ) show qualitative comparison.Given the results ofthe orbit integration we can plot thestreamfunction as a function of time by making use of Eq.( 78). We have done so n Fig. 19 a) for the point vortex casewhere E = 1 and t = 0, while Fig. 19(b) shows 4 for E = 1and t = 1, a later time ch osen so that 8=:~/2. Similarly, inFigs. 19(c) and 19(d) we plot 4 for the case where E = 0.6and t = 0 and t = ?, respectively (7 is chosen again so that8~=/2). Figure 19 should be compared to Fig. 15.In summary, we see hat there is qualitative agreement952 Phys. Fluids B, Vol. 3, No. 4, April 1991

between the simulation and the localized vortex model withincreasingl y better agreement for larger e.V. SUMMARY AND CONCLUSIONS

We have derived linear theoretical stability conditionsand growth rates for a plasma with shear flows, taking intoaccount gravity and magnetic shear. We analyzed the stabil-ity problem with a discontinuous background density andwith a smoothly varying density. The dispersion relationshows the presence of E XB shear flows can stabilize theinterchange and other related instabilities. The linear analy-sis of the gravitational instability shows that the interchange(R-T) mode is stabilized by theshear flow when the velocityshearu/a>fiorm[seeEqs. (18)and (48)J.Inthecase of the interchange mode, muc h shorter wavelengthmodes (much higher azimuthal modes) are destabilized.The mod e is strongly locahzed near the mode rational sur-face. The interchange mod e can be stabilized whenu/a 2 (g/L, ) /2.Implicit particle simulation results of the shear flow( K-H) instability resul ting from E X B drift in a magnet izedplasma show good agreement with the linear theory. Themaximum growth (0.2&a) and the threshold wave number( l/a) agree well with the observed growth rate and thresh-old. The linear K-H instability with k,, = 0 has a sharpboundary between the stable and unstable wave numbers,with the marginally stable modes having a local secular( - t) growt h arising, perhaps, from a ballistic resonance. ncontrast, three-di mensional (3-D) modes with k,! k < 8,hav e a redu ced growth rates. In t he simulation, if the tiltangle 6 2 0.02, the mod es become drift-wave-like and thegrowth rate is greatly reduced, while the unstable wave-length band expands. In the K-H instability, vortices thatgrow to a sufficient size trigger a secon dary nonlinea r insta-bility with smaller, subharmon ic wave numbers.The nonlinear instability is analyzed using a dynamicperiodic chain of localized vortex structures with an equilib-rium equival ent to the Kelvin-Stuart cats eye type solution,This equilibrium is observed to be unstable against the co-alescence nd tilt modes. The electrostatic energy increase ofa lower wave number mode (m = 1) (the growth rate ofm = 1 mode) is in reasonable agreement with the theory byPierrehumbert and Windal12 and the analysis of Sec. IV B.In a small-ampli tude regime, the tilt and coalescenc e nsta-bility and shear flow instability coexist. Even after the com-pletion of coalescence, he shear flow instability continues.In the caseof large amplitude, the tilt and coales cence nsta-bilities dominate the shear flow instability. Upon overshootof the tilt and coalescence, he coalesced vortices can againseparate nto two.The growth rate of the tilt angle is in goo d agreementwith the coal escen ce nstability ofa point vortex model in anappropriate range. The angle B increases aster than the ex-ponential function of time. The angl e B approache s rr andstays for a long time. The point vortex model can accuratelypredict the time profile of the rotation angl e of the vortices.The time scale prediction of the relative rotation of two vor-tices also provides a good explanation of the simulation.The results for the stability of the transitional layer of a

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t38

0-50 0 50

t

t8

t-I I

(c) (d)

FIG. 18. Nonlinear results ofthe localized vortex model . (a) 8(f) for E = 0.05,0.30,0.70, and 1 O. (b) &(t) for the same E values. (c) Plots of 6( I) that showtrajectories with initial conditions on the two sides of the separatrix. (d) &r(t) for two initial conditions, one just inside and the other just outside theseparatrix.

FIG. 19. Thestreamfunction for-thecases: (a) 4(q~,O;l), (b) qS(x,y,^t;l),(c) 4(xg,O;O.6), and (d) 4(x9&0.6). Here I and tare chosen so that therotation is approximately 180.953 Ws. Fluids B, Vol. 3, No. 4, April 1991

resistive plasma with a substantial change in density andperpendicular E X B flow velocity is given in terms of thetranscendental dispersion relation in Sec. II. The roots of thedispersion relations in the absence of shear flow describe theresistiveg instability and the collisional drift wave instabilitywith their different dependence on collisionality and mag-netic shear. In the presence of a sheared flow, the growthrates are strongly affected when the condition k,, Axu > yk issatisfied, where k,,, Ax, and rkY are the parameters of theinstability in the absence of the shear flow. The critical shearflow u obtained from this condition is shown in Table I.For the shear flows reported in the TEXT tokamak plas-ma with the new higher resolution probe measurements thecondition given previously is marginally satisfied so that weconclude that shear flow may have an influence on the edgeturbulence, even if it is not sufficiently strong to excite theK-H instability in that experiment.

There are two effects of the sheared E x B flow on theTajima et al. 953



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edge turbulence. Here, as in Ref. 1, we consider the directeffect of the shear flow on the wave dispersion relation,showing that the growth rate of the mode present in theabsence of shear flow can be strongly reduced. The secondaspect is that even for a given fixed level of backgroundwaves the transport across the magnetic field is reduced bythe shear flow. This decor relation effect of the transport hasbeen calculated by Shaing and Crume and Biglari et a[.using the ideas of relative diffusion or clump turbulence the-ory. A simpler estimate of the reduction of the transportcomes from consideri ng the single-particle motion of testparticles in a dominant fluctuation of mode number k andstrength &k; the radial excursion size is reduced from thedistance T./k, between nodes of the radial modes to the sizeAr = ( c$~/Bu) 2 given by the strength of the shear flow uand the amplitude of the potential fluctuation. Taylor et ai.,Shaing and Crume,9 Biglari et al., o and Burrell et al. arguethat the reduction in the plasma transport associated with Lto H mode transition occurs as a result of the increasedsheared flow velocity resulting from the deepeni ng of thenegative electrostatic potential well of the toroidal system.We show here how the increased strength of the shear flowchanges the stability conditions of the plasma.We consider the unstable sheared flow regime with im-plicit particle simulations and describe the resulting vortexdynamics by the motions of the vortex cores. To make ananalytic treatment of the vortex core dynamics, we idealizeto the case of point vortex dynamics, including the vortex-vortex interactions and the vortex-shear flow dynamics.This appears to give a good description of the principal pro-cessesof mutual rotations and coalescence of the vortices.Comparison wi th the simulations shows that a lowest-orderdescription of the turbulence follows from the vortex coredynamics wi th treating the density and pressure fields aspassively convected. The possibility of describing the rela-tionship of the density and potential fluctuations measuredin TEXT with the density being passively convected in theE x B flows given by the potential fluctuations has been pre-viously suggested by Bengtson and Rhodes.27Previous attempts to explain edge turbulence in TEXTby resistive hydrodynamic modes and by collisional modeshave not been completely successful. The theoretical formu-las used have neglected the effects of shear flow assumingthat the background velocity simply Doppl er shifted the fre-quencies of these instabilities. In view of the present theoryand new measurements of Ritz et a/.* (reportingdu,/dr

t. tajima et al- instabilities and vortex dynamics in shear flow of magnetized plasmas

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